Chapter 2 robot kinematics

Chapter 2Robot Kinematics: Position Analysis

2.1 INTRODUCTION

♦Forward Kinematics: to determine where the robot’s hand is? (If all joint variables are known)

♦Inverse Kinematics: to calculate what each joint variable is? (If we desire that the hand be located at a particular point)


2.2 ROBOTS AS MECHANISM

Fig. 2.1 A one-degree-of-freedom closed-loop four-bar mechanism

♦Multiple type robot have multiple DOF. (3 Dimensional, open loop, chain mechanisms)

Fig. 2.2 (a) Closed-loop versus (b) open-loop mechanism


2.3 MATRIX REPRESENTATION 2.3.1 Representation of a Point in Space

Fig. 2.3 Representation of a point in space

♦A point P in space : 3 coordinates relative to a reference frame

^^^

kcjbiaP zyx ++=


2.3 MATRIX REPRESENTATION 2.3.2 Representation of a Vector in Space

Fig. 2.4 Representation of a vector in space

♦A Vector P in space : 3 coordinates of its tail and of its head

^^^__

kcjbiaP zyx ++=

=

w

z

y

x

P__


2.3 MATRIX REPRESENTATION 2.3.3 Representation of a Frame at the Origin of a Fixed-Reference Frame

Fig. 2.5 Representation of a frame at the origin of the reference frame

♦Each Unit Vector is mutually perpendicular. : normal, orientation, approach vector

=

zzz

yyy

xxx

aon

aon

aon

F


2.3 MATRIX REPRESENTATION 2.3.4 Representation of a Frame in a Fixed Reference Frame

Fig. 2.6 Representation of a frame in a frame

♦Each Unit Vector is mutually perpendicular. : normal, orientation, approach vector

=

1000

zzzz

yyyy

xxxx

Paon

Paon

Paon

F


2.3 MATRIX REPRESENTATION 2.3.5 Representation of a Rigid Body

Fig. 2.8 Representation of an object in space

♦An object can be represented in space by attaching a frame to it and representing the frame in space.

=

1000

zzzz

yyyy

xxxx

objectPaon

Paon

Paon

F


2.4 HOMOGENEOUS TRANSFORMATION MATRICES

♦A transformation matrices must be in square form.

• It is much easier to calculate the inverse of square matrices.• To multiply two matrices, their dimensions must match.

=

1000

zzzz

yyyy

xxxx

Paon

Paon

Paon

F


2.5 REPRESENTATION OF TRANSFORMATINS 2.5.1 Representation of a Pure Translation

Fig. 2.9 Representation of an pure translation in space

♦A transformation is defined as making a movement in space. • A pure translation.• A pure rotation about an axis.• A combination of translation or rotations.

=

1000

100

010

001

z

y

x

d

d

d

T


2.5 REPRESENTATION OF TRANSFORMATINS 2.5.2 Representation of a Pure Rotation about an Axis

Fig. 2.10 Coordinates of a point in a rotating frame before and after rotation.

♦Assumption : The frame is at the origin of the reference frame and parallel to it.

Fig. 2.11 Coordinates of a point relative to the reference frame and rotating frame as viewed from the x-axis.


2.5 REPRESENTATION OF TRANSFORMATINS 2.5.3 Representation of Combined Transformations

Fig. 2.13 Effects of three successive transformations

♦A number of successive translations and rotations….

Fig. 2.14 Changing the order of transformations will change the final result


2.5 REPRESENTATION OF TRANSFORMATINS 2.5.5 Transformations Relative to the Rotating Frame

Fig. 2.15 Transformations relative to the current frames.

♦Example 2.8


2.6 INVERSE OF TRANSFORMATION MATIRICES

Fig. 2.16 The Universe, robot, hand, part, and end effecter frames.

♦Inverse of a matrix calculation steps : • Calculate the determinant of the matrix. • Transpose the matrix. • Replace each element of the transposed matrix by its own minor(adjoint matrix). • Divide the converted matrix by the determinant.


2.7 FORWARD AND INVERSE KINEMATICS OF ROBOTS

Fig. 2.17 The hand frame of the robot relative to the reference frame.

♦Forward Kinematics Analysis: • Calculating the position and orientation of the hand of the robot. • If all robot joint variables are known, one can calculate where the robot is at any instant. • Recall Chapter 1.


2.7 FORWARD AND INVERSE KINEMATICS OF ROBOTS 2.7.1 Forward and Inverse Kinematics Equations for Position

♦Forward Kinematics and Inverse Kinematics equation for position analysis : (a) Cartesian (gantry, rectangular) coordinates. (b) Cylindrical coordinates. (c) Spherical coordinates. (d) Articulated (anthropomorphic, or all-revolute) coordinates.


2.7 FORWARD AND INVERSE KINEMATICS OF ROBOTS 2.7.1 Forward and Inverse Kinematics Equations for Position 2.7.1(a) Cartesian (Gantry, Rectangular) Coordinates

♦IBM 7565 robot • All actuator is linear. • A gantry robot is a Cartesian robot.

Fig. 2.18 Cartesian Coordinates.

==

1000

100

010

001

z

y

x

cartPR

P

P

P

TT


2.7 FORWARD AND INVERSE KINEMATICS OF ROBOTS 2.7.1 Forward and Inverse Kinematics Equations for Position 2.7.1(b) Cylindrical Coordinates

♦2 Linear translations and 1 rotation • translation of r along the x-axis • rotation of α about the z-axis • translation of l along the z-axis

Fig. 2.19 Cylindrical Coordinates.

−

==

1000

100

0

0

l

rSCS

rCSC

TT cylPR ααα

ααα

,0,0))Trans(,)Rot(Trans(0,0,),,( rzllrTT cylPR αα ==


2.7 FORWARD AND INVERSE KINEMATICS OF ROBOTS 2.7.1 Forward and Inverse Kinematics Equations for Position 2.7.1(c) Spherical Coordinates

♦2 Linear translations and 1 rotation • translation of r along the z-axis • rotation of β about the y-axis • rotation of γ along the z-axis

Fig. 2.20 Spherical Coordinates.

−⋅⋅⋅⋅⋅−⋅

==

1000

0 βββγβγβγγβγβγβγγβ

rCCS

SrSSSCSC

CrSCSSCC

TT sphPR

))Trans()Rot(Rot()( 0,0,,,,, γβγβ yzlrsphPR TT ==


2.7 FORWARD AND INVERSE KINEMATICS OF ROBOTS 2.7.1 Forward and Inverse Kinematics Equations for Position 2.7.1(d) Articulated Coordinates

♦3 rotations -> Denavit-Hartenberg representation

Fig. 2.21 Articulated Coordinates.


2.7 FORWARD AND INVERSE KINEMATICS OF ROBOTS 2.7.2 Forward and Inverse Kinematics Equations for Orientation

♦ Roll, Pitch, Yaw (RPY) angles♦ Euler angles♦ Articulated joints


2.7 FORWARD AND INVERSE KINEMATICS OF ROBOTS 2.7.2 Forward and Inverse Kinematics Equations for Orientation 2.7.2(a) Roll, Pitch, Yaw(RPY) Angles

♦Roll: Rotation of about -axis (z-axis of the moving frame)♦Pitch: Rotation of about -axis (y-axis of the moving frame)♦Yaw: Rotation of about -axis (x-axis of the moving frame)

aaφoφnφ

on

Fig. 2.22 RPY rotations about the current axes.


2.7 FORWARD AND INVERSE KINEMATICS OF ROBOTS 2.7.2 Forward and Inverse Kinematics Equations for Orientation 2.7.2(b) Euler Angles

Fig. 2.24 Euler rotations about the current axes.

♦Rotation of about -axis (z-axis of the moving frame) followed by♦Rotation of about -axis (y-axis of the moving frame) followed by♦Rotation of about -axis (z-axis of the moving frame).

aφθψ

oa


2.7 FORWARD AND INVERSE KINEMATICS OF ROBOTS 2.7.2 Forward and Inverse Kinematics Equations for Orientation 2.7.2(c) Articulated Joints

Consult again section 2.7.1(d)…….


2.7 FORWARD AND INVERSE KINEMATICS OF ROBOTS 2.7.3 Forward and Inverse Kinematics Equations for Orientation

)()( ,,,, noazyxcartHR RPYPPPTT φφφ×=

)()( ,,,, ψθγβ φEulerTT rsphHR ×=

♦ Assumption : Robot is made of a Cartesian and an RPY set of joints.

♦ Assumption : Robot is made of a Spherical Coordinate and an Euler angle.

Another Combination can be possible……

Denavit-Hartenberg Representation


2.8 DENAVIT-HARTENBERG REPRESENTATION OF FORWARD KINEMATIC EQUATIONS OF ROBOT

• Denavit-Hartenberg Representation :

Fig. 2.25 A D-H representation of a general-purpose joint-link combination

@ Simple way of modeling robot links and joints for any robot configuration, regardless of its sequence or complexity.

@ Transformations in any coordinates is possible.

@ Any possible combinations of joints and links and all-revolute articulated robots can be represented.



• Denavit-Hartenberg Representation procedures:

Start point:

Assign joint number n to the first shown joint.

Assign a local reference frame for each and every joint before or

after these joints.

Y-axis does not used in D-H representation.



• Procedures for assigning a local reference frame to each joint:

.All joints are represented by a z-axis ٭ (right-hand rule for rotational joint, linear movement for prismatic joint)

The common normal is one line mutually perpendicular to any two ٭

skew lines.

.Parallel z-axes joints make a infinite number of common normal ٭

Intersecting z-axes of two successive joints make no common ٭

normal between them(Length is 0.).



• Symbol Terminologies :

⊙ θ : A rotation about the z-axis.

⊙ d : The distance on the z-axis.

⊙ a : The length of each common normal (Joint offset).

⊙ α : The angle between two successive z-axes (Joint twist)

Only θ and d are joint variables.



• The necessary motions to transform from one reference frame to the next.

(I) Rotate about the zn-axis an able of θn+1. (Coplanar)

(II) Translate along zn-axis a distance of dn+1 to make xn and xn+1

colinear.

(III) Translate along the xn-axis a distance of an+1 to bring the origins

of xn+1 together.

(IV) Rotate zn-axis about xn+1 axis an angle of αn+1 to align zn-axis

with zn+1-axis.


2.9 THE INVERSE KINEMATIC SOLUTION OF ROBOT

• Determine the value of each joint to place the arm at a desired position and orientation.

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+++−+

++−−

−−+

−

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−−−

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=

1000

)()()()(

)()()()(

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22323423415152341651

6234652341

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aSaSaSSSCCCCSSCCCS

aCaCaCSCCSCSCSC

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aCaCaCCCSSCCCSS

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=

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zzzz

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xxxx

paon

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6543211

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paon

paon

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yyyy

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==

× −−

6543211

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AAAAApaon

paon

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zzzz

yyyy

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=

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= −

x

y

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p11 tanθ

)())((

)())((tan

423433423411233

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aSPaSaCSpCpaaC

aCSpCpaSaSpaaC

zyx

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−+−++−+−−+= −θ

= −

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313 tan

C

Sθ

322344 θθθθ −−=

yx

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11

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)(tan

−++= −θ

zyx

zyx

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)(

)(tan

++−++−= −θ


2.10 INVERSE KINEMATIC PROGRAM OF ROBOTS• A robot has a predictable path on a straight line, • Or an unpredictable path on a straight line.

.A predictable path is necessary to recalculate joint variables ٭

(Between 50 to 200 times a second)

To make the robot follow a straight line, it is necessary to break ٭

the line into many small sections.

.All unnecessary computations should be eliminated ٭

Fig. 2.30 Small sections of movement for straight-line motions


2.11 DEGENERACY AND DEXTERITY

∴Degeneracy : The robot looses a degree of freedom and thus cannot perform as desired.

,When the robot’s joints reach their physical limits ٭ and as a result, cannot move any further.

In the middle point of its workspace if the z-axes ٭ of two similar joints becomes colinear.

Fig. 2.31 An example of a robot in a degenerate position.

∴Dexterity : The volume of points where one can position the robot as desired, but not orientate it.


2.12 THE FUNDAMENTAL PROBLEM WITH D-H REPRESENTATION

∴Defect of D-H presentation : D-H cannot represent any motion about the y-axis, because all motions are about the x- and z-axis.

Fig. 2.31 The frames of the Stanford Arm.

# θ d a α

1 θ1 0 0 -90

2 θ2 d1 0 90

3 0 d1 0 0

4 θ4 0 0 -90

5 θ5 0 0 90

6 θ6 0 0 0

TABLE 2.3 THE PARAMETERS TABLE FOR THE STANFORD ARM

Chapter 2 robot kinematics

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